Some applications of extriangulated categories

TL;DR

This survey explores three applications of extriangulated categories across homotopical algebra, combinatorics, and representation theory, highlighting their versatility in constructing model structures, polytopal realizations, and mutation frameworks.

Contribution

It introduces novel applications of extriangulated categories in model structures, polytopal fan realizations, and mutation theories, expanding their utility in various mathematical fields.

Findings

Extriangulated categories facilitate model category structures with triangulated homotopy categories.

They enable the construction of polytopal realizations of g-vector fans, generalizing ABHY's scattering amplitudes.

Provide a framework for mutation theories in hereditary extriangulated categories, unifying various concepts in representation theory and combinatorics.

Abstract

Extriangulated categories axiomatize extension-closed subcategories of triangulated categories and generalise both exact categories and triangulated categories. This survey article presents three applications of extriangulated categories to homotopical algebra, algebraic combinatorics and representation theory. The first shows that, via some generalised Hovey's correspondence, extriangulated categories easily give rise to model category structures with triangulated homotopy categories. As a second application, extriangulated structures play a fondamental role in the construction of polytopal realisations of $g$-vector fans. This allows for a generalisation of ABHY's construction appearing in the study of scattering amplitudes in theoretical physics. Lastly, extriangulated categories provide a convenient framework for studying mutations in representation theory and flips in algebraic combinatorics. In nice enough hereditary extriangulated categories, there is a well-behaved theory of mutation for silting objects, which encompass cluster tilting, two-term silting, relative tilting, mutation of maximal almost-rigid modules, flip of dissections and mutation of intermediate co-$t$-structures.